Irreducible modular representations of GL2 of a local field

The image of Colmez’s Montreal functor

“…To prove this, one has to compute all the Ext 1 groups between irreducible representations of G, which have been classified by Barthel-Livne [1] and Breuil [16]. In many cases these computations have been dealt with by Breuil and the author [20], Colmez [23] and Emerton [32] by different methods, and the computation was completed in [56].…”

“…In Theorem 3.39 we devise a criterion for commutativity of E. Further, suppose that for every exact sequence (1) 0 → Q ⊕r → T → Q → 0 with dim Hom C(k) (T, S) = 1 we have dim Ext 1 C(k) (T, S) ≤ r(r−1)…”

“…If G = GL 2 (Q p ) then it follows from the classification in [1] and [16] that every smooth irreducible k-representation of G with a central character is admissible and hence any smooth finite length k-representation of G with a central character is admissible. So the assumption that Q is finitely generated over O[[H]] will be automatically satisfied.…”

Explicit Reduction Modulo p of Certain Two-Dimensional Crystalline Representations

“…The classification is due to Barthel-Livne and Breuil, and is as follows (see for example Théorème 2.7.1 of [Bre03a], which summarises results in [BL94], and also Corollaire 4.1.1, Corollaire 4.1.4 and Corollaire 4.1.5 of loc. cit.…”

On unitary deformations of smooth modular representations

“…This heavily builds upon results of M. Emerton concerning his functor of 'ordinary parts' ( [Eme10b]). We also remark that the structure of such principal series representations ρ over k has been made explicit by the work of Barthel/Livné ( [BL94]). …”